Transport properties of continuum systems near the percolation threshold

In-vivo blood pressure sensing with bi-filler nanocomposite

Conductive elastomers present desirable qualities for sensing pressure in-vivo, such as high piezoresistance in tiny volumes, conformability and, biocompatibility. Many electrically conductive nanocomposites however, are susceptible to electrical drift following repeated stress cycles and chemical aging. Here we propose an innovative approach to stabilize nanocomposite percolation network against incomplete recovery to improve reproducibility and facilitate sensor calibration. We decouple the tunnelling-percolation network of highly-oriented pyrolytic graphite (HOPG) nanoparticles from the incomplete viscoelastic recovery of the polydimethylsiloxane (PDMS) matrix by inserting minute amounts of insulating SiO2 nanospheres. SiO2 nanospheres effectively reduce the number of nearest neighbours at each percolation node switching off the parallel electrical pathways that might become activated under incomplete viscoelastic relaxation. We varied the size of SiO2 nanospheres and their filling fraction to demonstrate nearly complete piezoresistance recovery when SiO2 and HOPG nanoparticles have equal diameters (≈400 nm) and SiO2 and HOPG volume fractions are 1 % and 29.5 % respectively. We demonstrate an in-vivo blood pressure sensor based on this bi-filler composite.

Percolation and conductivity in evolving disordered media

Percolation theory and the associated conductance networks have provided deep insights into the flow and transport properties of a vast number of heterogeneous materials and media. In practically all cases, however, the conductance of the networks' bonds remains constant throughout the entire process. There are, however, many important problems in which the conductance of the bonds evolves over time and does not remain constant. Examples include clogging, dissolution and precipitation, and catalytic processes in porous materials, as well as the deformation of a porous medium by applying an external pressure or stress to it that reduces the size of its pores. We introduce two percolation models to study the evolution of the conductivity of such networks. The two models are related to natural and industrial processes involving clogging, precipitation, and dissolution processes in porous media and materials. The effective conductivity of the models is shown to follow known power laws near the percolation threshold, despite radically different behavior both away from and even close to the percolation threshold. The behavior of the networks close to the percolation threshold is described by critical exponents, yielding bounds for traditional percolation exponents. We show that one of the two models belongs to the traditional universality class of percolation conductivity, while the second model yields nonuniversal scaling exponents.

Universal scaling for the permeability of random packs of overlapping and nonoverlapping particles

Constraining fluid permeability in porous media is central to a wide range of theoretical, industrial, and natural processes. In this Letter, we validate a scaling for fluid permeability in random and lattice packs of spheres and show that the permeability of packs of both hard and overlapping spheres of any sphere size or size distribution collapse to a universal curve across all porosity ϕ in the range of ϕ_{c}<ϕ<1, where ϕ_{c} is the percolation threshold. We use this universality to demonstrate that permeability can be predicted using percolation theory at ϕ_{c}<ϕ≲0.30, Kozeny-Carman models at 0.30≲ϕ≲0.40, and dilute expansions of Stokes theory for lattice models at ϕ≳0.40. This result leads us to conclude that the inverse specific surface area, rather than an effective sphere size or pore size is a universal controlling length scale for hydraulic properties of packs of spheres. Finally, we extend this result to predict the permeability for some packs of concave nonspherical particles.

Scale dependence of tortuosity and diffusion: Finite-size scaling analysis

Physical and hydraulic properties of porous media are routinely measured/simulated at smaller scales e.g., pore and core. However, their determination at larger scales e.g., field and reservoir has still been a great challenge. Although understanding the scale dependence of transport modes in rocks and soils is essential, the porous media community still lacks in a solid theoretic framework. In this short communication, we propose finite-size scaling analysis from physics to investigate the scale dependence of tortuosity and diffusion coefficient. By comparing with two- and three-dimensional simulations, we demonstrate that the finite-size scaling analysis is a powerful approach. More specifically, we show that the plot of simulated tortuosity or diffusion coefficient versus scale looks scattered. However, after applying the finite-size scaling analysis, the data collapse together showing a quasi-universal trend.

Simple holistic solution to Archie's-law puzzle in porous media

In this paper, we account for the many critical exponents derived from the studies of the electrical conductivity in porous media by applying analysis of the well-known relation known as Archie's law. In spite of its seeming simplicity this law is considered to be "poorly understood," and the question that was and still is debated in the literature is whether there is some "hidden physics" in this law, or if it is "strictly a parametrization use for curve fitting with a priori no physical meaning." Our solution to the corresponding long-debated 78 years old puzzle is based on the classical percolation theory, but it also involves a principle that is based on continuum percolation. This principle is that the electrical properties of a percolation system are determined by the interplay between the connectivity of the conducting objects in that system, and the connectivity of the intersections between pairs of them. We thus propose a general concept that we call an electrically affected connectivity, and we predict the corresponding evolvement of the conductivity critical exponent with the increase of the content of the electrically conducting phase. Then, we show that the zerolike threshold that characterizes Archie's law is what enables the observation of this evolution. Combining the above principle and the latter feature, we provide a holistic, yet simple, solution to the longstanding controversy surrounding this law and its practical applications. In contrast with many previous claims that Archie's law lacks a physical basis, and the commonly suggested experiential explanations for it, we provide a solution that is physically based and thus elucidates Archie's law by showing clearly that it represents a bona fide phase transition phenomenon. This conclusion and its generality are strongly supported by the fact that it also explains the behavior of the electrical conductivity exponents in nonporous systems such as composite materials. The predicted ability to extract the long sought microgeometrical information from Archie's-law data, within the framework of the percolation phase transition, is expected to open a new direction in the understanding and the applications of this law.

Permeability of packs of polydisperse hard spheres

The permeability of packs of spheres is important in a wide range of physical scenarios. Here, we create numerically generated random periodic domains of spheres that are polydisperse in size and use lattice-Boltzmann simulations of fluid flow to determine the permeability of the pore phase interstitial to the spheres. We control the polydispersivity of the sphere size distribution and the porosity across the full range from high porosity to a close packing of spheres. We find that all results scale with a Stokes permeability adapted for polydisperse sphere sizes. We show that our determination of the permeability of random distributions of spheres is well approximated by models for cubic arrays of spheres at porosities greater than ∼0.38, without any fitting parameters. Below this value, the Kozeny-Carman relationship provides a good approximation for dense, closely packed sphere packs across all polydispersivity.

Ant collective cognition allows for efficient navigation through disordered environments

The cognitive abilities of biological organisms only make sense in the context of their environment. Here, we study longhorn crazy ant collective navigation skills within the context of a semi-natural, randomized environment. Mapping this biological setting into the ‘Ant-in-a-Labyrinth’ framework which studies physical transport through disordered media allows us to formulate precise links between the statistics of environmental challenges and the ants’ collective navigation abilities. We show that, in this environment, the ants use their numbers to collectively extend their sensing range. Although this extension is moderate, it nevertheless allows for extremely fast traversal times that overshadow known physical solutions to the ‘Ant-in-a-Labyrinth’ problem. To explain this large payoff, we use percolation theory and prove that whenever the labyrinth is solvable, a logarithmically small sensing range suffices for extreme speedup. Overall, our work demonstrates the potential advantages of group living and collective cognition in increasing a species’ habitable range.

Nanocomposite Polymeric Materials Based on Eucalyptus Lignoboost® Kraft Lignin for Liquid Sensing Applications

This study reports the synthesis of polyurethane–lignin copolymer blended with carbon multilayer nanotubes to be used in all-solid-state potentiometric chemical sensors. Known applicability of lignin-based polyurethanes doped with carbon nanotubes for chemical sensing was extended to eucalyptus LignoBoost^® kraft lignin containing increased amounts of polyphenolic groups from concomitant tannins that were expected to impart specificity and sensitivity to the sensing material. Synthesized polymers were characterized using FT-MIR spectroscopy, electrical impedance spectroscopy, scanning electron microscopy, thermogravimetric analysis, and differential scanning calorimetry and are used for manufacturing of all solid-state potentiometric sensors. Potentiometric sensor with LignoBoost^® kraft lignin-based polyurethane membrane displayed theoretical response and high selectivity to Cu (II) ions, as well as long-term stability.

Modeling the electrical resistivity of polymer composites with segregated structures

Hybrid carbon nanotube composites with two different types of fillers have attracted considerable attention for various advantages. The incorporation of micro-scale secondary fillers creates an excluded volume that leads to the increase in the electrical conductivity. By contrast, nano-scale secondary fillers shows a conflicting behavior of the decreased electrical conductivity with micro-scale secondary fillers. Although several attempts have been made in theoretical modeling of secondary-filler composites, the knowledge about how the electrical conductivity depends on the dimension of secondary fillers was not fully understood. This work aims at comprehensive understanding of the size effect of secondary particulate fillers on the electrical conductivity, via the combination of Voronoi geometry induced from Swiss cheese models and the underlying percolation theory. This indicates a transition in the impact of the excluded volume, i.e., the adjustment of the electrical conductivity was measured in cooperation with loading of second fillers with different sizes.

Fluid flow through packings of elastic shells

Fluid transport in porous materials is commonly studied in geological samples (soil, sediments, etc.) or idealized systems, but the fluid flow through compacted granular materials, consisting of substantially strained granules, remains relatively unexplored. As a step toward filling this gap, we study a model of liquid transport in packings of deformable elastic shells using finite-element and lattice-Boltzmann methods. We find that the fluid flow abruptly vanishes as the porosity of the material falls below a critical value, and the flow obstruction exhibits features of a percolation transition. We further show that the fluid flow can be captured by a simplified permeability model in which the complex porous material is replaced by a collection of disordered capillaries, which are distributed and shaped by the percolation transition. To that end, we numerically explore the divergence of hydraulic tortuosity τ_{H} and the decrease of a hydraulic radius R_{h} as the percolation threshold is approached. We interpret our results in terms of scaling predictions derived from the percolation theory applied to random packings of spheres.

Topological inversions in coalescing granular media control fluid-flow regimes

Sintering-or coalescence-of viscous droplets is an essential process in many natural and industrial scenarios. Current physical models of the dynamics of sintering are limited by the lack of an explicit account of the evolution of microstructural geometry. Here, we use high-speed time-resolved x-ray tomography to image the evolving geometry of a sintering system of viscous droplets, and use lattice Boltzmann simulations of creeping fluid flow through the reconstructed pore space to determine its permeability. We identify and characterize a topological inversion, from spherical droplets in a continuous interstitial gas, to isolated bubbles in a continuous liquid. We find that the topological inversion is associated with a transition in permeability-porosity behavior, from Stokes permeability at high porosity, to percolation theory at low porosity. We use these findings to construct a unified physical description that reconciles previously incompatible models for the evolution of porosity and permeability during sintering.

Unified Model for Pseudononuniversal Behavior of the Electrical Conductivity in Percolation Systems

Many values of the observed conductivity percolation exponent t cannot be explained by the classical universal theory or by the existing nonuniversal theories. In particular, the 1.3≤t≤4.0 clustering of t values, in both composite materials and porous media has not been accounted for. In this work we were concerned with a pseudononuniversal percolation behavior that, unlike the genuine nonuniversal behavior, explains the statistics of the experimentally observed percolation conductivity exponents in continuum systems.

Possible origin of the smaller-than-universal percolation-conductivity exponent in the continuum

For quite a few systems in the continuum, such as carbon nanotube polymer composites and segregated composites, percolation electrical conductivity exponents that are much smaller than the universal value have been reported. This is unexpected in view of the classical lattice percolation theory. Here we provide a simple general phenomenological model that accounts for such observations within the framework of universality. We suggest that these small value exponents are due to the interplay between the connectivity and the structural variations that follow the increase of the fractional volume content of the conducting phase.

Neuromorphic behavior in percolating nanoparticle films

We show that the complex connectivity of percolating networks of nanoparticles provides a natural solid-state system in which bottom-up assembly provides a route to realization of neuromorphic behavior. Below the percolation threshold the networks comprise groups of particles separated by tunnel gaps; an applied voltage causes atomic scale wires to form in the gaps, and we show that the avalanche of switching events that occurs is similar to potentiation in biological neural systems. We characterize the level of potentiation in the percolating system as a function of the surface coverage of nanoparticles and other experimentally relevant variables, and compare our results with those from biological systems. The complex percolating structure and the electric field driven switching mechanism provide several potential advantages in comparison to previously reported solid-state neuromorphic systems.

Conductivity scaling in supercritical percolation of nanoparticles--not a power law

The power-law behavior widely observed in supercritical percolation systems of conductive nanoparticles may merely be a phenomenological approximation to the true scaling law not yet discovered. In this work, we derive a comprehensive yet simple scaling law and verify its extensive applicability to various experimental and numerical systems. In contrast to the power law which lacks theoretical backing, the new scaling law is explanatory and predictive, and thereby helpful to gain more new insights into percolation systems of conductive nanoparticles.

A scalable and facile synthesis of alumina/exfoliated graphite composites by attrition milling

We present a facile one-pot synthesis of alumina/exfoliated graphite composite having excellent electrical conductivity (>1,000 S m⁻¹), fracture toughness (5.6 MPa m^0.5), and wear resistance, which is enhanced by 7.7 times compared to pure alumina.

Does hindered transport theory apply to desalination membranes?

As reverse osmosis (RO) and nanofiltration polyamide membranes become increasingly used for water purification, prediction of pollutant transport is required for membrane development and process engineering. Many popular models use hindered transport theory (HTT), which considers a spherical solute moving through an array of fluid-filled rigid cylindrical pores. Experiments and molecular dynamic simulations, however, reveal that polyamide membranes have a distinctly different structure of a "molecular sponge", a network of randomly connected voids widely distributed in size. In view of this disagreement, this study critically examined the validity of HTT by directly measuring diffusivities of several alcohols within a polyamide film of commercial RO membrane using attenuated total reflection-FTIR. It is found that measured diffusivities deviate from HTT predictions by as much as 2-3 orders of magnitude. This result indicates that HTT does not adequately describe solute transport in desalination membranes. As a more adequate alternative, the concept of random resistor networks is suggested, with resistances described by models of activated transport in "soft" polymers without a sharp size cutoff and with a proper address of solute partitioning.

Percolation model for a selective response of the resistance of composite semiconducting np systems with respect to reducing gases

It is shown that a two-component percolation model on a simple cubic lattice can explain an experimentally observed behavior [Savage et al., Sens. Actuators B 79, 17 (2001); Sens. Actuators B 72, 239 (2001).], namely, that a network built up by a mixture of sintered nanocrystalline semiconducting n and p grains can exhibit selective behavior, i.e., respond with a resistance increase when exposed to a reducing gas A and with a resistance decrease in response to another reducing gas B. To this end, a simple model is developed, where the n and p grains are simulated by overlapping spheres, based on realistic assumptions about the gas reactions on the grain surfaces. The resistance is calculated by random walk simulations with nn, pp, and np bonds between the grains, and the results are found in very good agreement with the experiments. Contrary to former assumptions, the np bonds are crucial to obtain this accordance.

Dispersion in porous media, continuous-time random walks, and percolation

A promising approach to the modeling of anomalous (non-Gaussian) dispersion in flow through heterogeneous porous media is the continuous-time random walk (CTRW) method. In such a formula on the waiting time distribution ψ(t) is usually assumed to be given by ψ(t)∼t-1-α, with α fitted to the experimental data. The exponent α is also related to the power-law growth of the mean-square displacement of the solute with the time t <R2(t)> ∼ tζ. Invoking percolation and using a scaling analysis, we relate α to the geometrical exponents of percolation (ν, β, and βB) as well as the exponents μ and e that characterize the power-law behavior of the effective conductivity and permeability of porous media near the percolation threshold. We then explain the cause of the nonuniversality of α in terms of the nonuniversality of μ and e in continuum systems, and in percolation models with long-range correlations, and propose bounds for it. The results are consistent with the experimental data, both at the laboratory and field scales.

Permeability, porosity, and percolation properties of two-dimensional disordered fracture networks

Using extensive Monte Carlo simulations, we study the effective permeability, porosity, and percolation properties of two-dimensional fracture networks in which the fractures are represented by rectangles of finite widths. The parameters of the study are the width of the fractures and their number density. For low and intermediate densities, the average porosity of the network follows a power-law relation with the density. The exponent of the power law itself depends on the fractures' width through a power law. For an intermediate range of the densities, the effective permeability scales with the fractures' width as a power law, with an exponent that depends on the density. For high densities the effective permeability also depends on the porosity through a power law, with an exponent that depends on the fractures' width. In agreement with the results, experimental data also indicate the existence of a power-law relationship between the effective permeability and porosity in consolidated sandstones and sedimentary rocks with a nonuniversal exponent. The percolation threshold or critical number density of the fractures depends on their width and is maximum if they are represented by squares, rather than rectangles.

Two-dimensional continuum percolation threshold for diffusing particles of nonzero radius

Lateral diffusion in the plasma membrane is obstructed by proteins bound to the cytoskeleton. The most important parameter describing obstructed diffusion is the percolation threshold. The thresholds are well known for point tracers, but for tracers of nonzero radius, the threshold depends on the excluded area, not just the obstacle concentration. Here thresholds are obtained for circular obstacles on the continuum. Random obstacle configurations are generated by Brownian dynamics or Monte Carlo methods, the obstacles are immobilized, and the percolation threshold is obtained by solving a bond percolation problem on the Voronoi diagram of the obstacles. The percolation threshold is expressed as the diameter of the largest tracer that can cross a set of immobile obstacles at a prescribed number density. For random overlapping obstacles, the results agree with the known analytical solution quantitatively. When the obstacles are soft disks with a 1/r(12) repulsion, the percolating diameter is approximately 20% lower than for overlapping obstacles. A percolation model predicts that the threshold is highly sensitive to the tracer radius. To our knowledge, such a strong dependence has so far not been reported for the plasma membrane, suggesting that percolation is not the factor controlling lateral diffusion. A definitive experiment is proposed.

Effective transport properties of random composites: continuum calculations versus mapping to a network

The effective transport properties and percolation of continuum composites have commonly been studied using discrete models, i.e., by mapping the continuum to a lattice or network. In this study we instead directly solve the continuum transport equations for composite microstructures both analytically and numerically, and we extract the continuum percolation threshold and scaling exponents for the two-dimensional square tile system. We especially focus on the role of corner contacts on flux flow and further show that mapping such "random checkerboard" systems to a network leads to a spurious secondary percolation threshold and causes shifts in the critical scaling exponents of the effective transport properties.

Direct measurement of the percolation probability in carbon nanofiber-polyimide nanocomposites

We present the first experimental measurement of the geometric critical exponent beta associated with the percolation probability, the probability a metallic filler belongs to the conducting network, of an electrical composite. The technique employs conducting-tip atomic force microscopy to obtain a conducting areal density, and is demonstrated on polyimide nanocomposites containing different concentrations of carbon nanofibers. We find beta approximately 1 and t (the exponent for bulk conductivity) approximately 3. These values are consistent with the predictions for the Bethe lattice and larger than the values predicted in the 3D lattice percolation model. Hence, this electrical composite likely belongs to the same universality class as the Bethe lattice. The ability to measure geometric and transport critical exponents on the same material is critical to drawing this conclusion.

Optimal percolation of disordered segregated composites

We evaluate the percolation threshold values for a realistic model of continuum segregated systems, where random spherical inclusions forbid the percolating objects, modeled by hardcore spherical particles surrounded by penetrable shells, to occupy large regions inside the composite. We find that the percolation threshold is generally a nonmonotonous function of segregation, and that an optimal (i.e., minimum) critical concentration exists well before maximum segregation is reached. We interpret this feature as originating from a competition between reduced available volume effects and enhanced concentrations needed to ensure percolation in the highly segregated regime. The relevance with existing segregated materials is discussed.

A comparison of viscosity-concentration relationships for emulsions

Differential effective medium theory (D-EMT) has been used by a number of investigators to derive expressions for the shear viscosity of a colloidal suspension or an emulsion as a function of the volume fraction of the dispersed phase. Pal and Rhodes [R. Pal, E. Rhodes, J. Rheol. 33 (7) (1989) 1021-1045] used D-EMT to derive a viscosity-concentration expression for non-Newtonian emulsions, in which variations among different oil-water emulsions were accommodated by fitting the value of an empirical solvation factor by matching the volume fraction at which the ratio of each emulsion was experimentally observed to have a viscosity 100 times greater than that of the pure solvent. When the particles in suspension have occluded volume due to solvation or flocculation, we show that the application of D-EMT to the problem becomes more ambiguous than these investigators have indicated. In addition, the resulting equations either do not account for the limiting behavior near the critical concentration, that is, the concentration at which the viscosity diverges, or they incorporate this critical behavior in an ad hoc way. We suggest an alternative viscosity-concentration equation for emulsions, based on work by Bicerano and coworkers [J. Bicerano, J.F. Douglas, D.A. Brune, J. Macromol. Sci., Rev. Macromol. Chem. Phys. C 39 (4) (1999) 561-642]. This alternative equation has the advantages that (1) its parameters are more closely related to physical properties of the suspension and (2) it recovers the correct limiting behavior both in the dilute limit and near the critical concentration for rigid particles. In addition, the equation can account for the deformability of flexible particles in the semidilute regime. The proposed equation is compared to the equation proposed by Pal and Rhodes.

Self-healing materials: a review

The ability of materials to self-heal from mechanical and thermally induced damage is explored in this paper and has significance in the field of fracture and fatigue. The history and evolution of several self-repair systems is examined including nano-beam healing elements, passive self-healing, autonomic self-healing and ballistic self-repair. Self-healing mechanisms utilized in the design of these unusual materials draw much information from the related field of polymer-polymer interfaces and crack healing. The relationship of material damage to material healing is examined in a manner to provide an understanding of the kinetics and damage reversal processes necessary to impart self-healing characteristics. In self-healing systems, there are transitions from hard-to-soft matter in ballistic impact and solvent bonding and conversely, soft-to-hard matter transitions in high rate yielding materials and shear-thickening fluids. These transitions are examined in terms of a new theory of the glass transition and yielding, viz., the twinkling fractal theory of the hard-to-soft matter transition. Success in the design of self-healing materials has important consequences for material safety, product performance and enhanced fatigue lifetime.